All posts for the month September, 2017

(All rights belong to their owners. Images used here for review purposes only.)

The Moscow Puzzles, by Boris A. Kordemsky, Edited by Martin Gardner (320 pages)
This is one more book I received for my birthday, and is also related to Martin Gardner. Kordemsky (1907-1999) was a Russian high school teacher, and math and science writer. He wrote over 70 books and popular math articles, making him one of the more prolific Russian writers of recreational mathematics. The above book specifically represents a collection of 359 math puzzles, published in 1956. It was eventually translated into English by Dr. Albert Parry, then cleaned up, with units and references converted to feet, inches, dollars, etc., where allowed by the puzzle, by Martin Gardner. My copy states that it was published in 1971 by Charles Scribner’s Sons, but the amazon link says that this book came out June, 2017, from PMA publishing. I’m assuming it’s a reprint, but the copyright information on the flyleaf is confusing. Anyway, many of the puzzles here had also been made popular by Henry Dudeney and Sam Loyd, but Kordemsky put his own twists on them.

Martin states that in editing the puzzles he wanted to maintain their Russian feel as much as possible, and I think he’d succeeded. In fact, I really like the way the set-ups of the individual puzzles work, with references to steam locomotives, dacha, worker’s collectives and lathe workers – there’s a strong sense of someone living in the 1950’s Soviet Union talking about the people around them.

When I was in high school in the 70’s, my math teacher tried to get the class to learn how to solve problems on our own by posting similar problems on a flip chart tablet, and only telling us if our answers were right or not. Some of those problems mimic the ones in the Moscow Problems book, but I was never able to figure them out myself – I just wanted the solutions to be handed to me.

Going through this book now, I’m getting about 1/3 of the problems correct, but I’m not actively sitting down with a sheet of paper and a pencil to solve everyone of them on my own. I’m more interested in the theory behind the puzzles. However, there are two problems in specific that I really like – the impossible dovetail joint, and “a swimmer and a hat.” I’ve seen pictures of the dovetail joint on the net before, and I like seeing it showing up here in the book along with the solution. But, what makes the “swimmer” puzzle fascinating is that it’s a variation on George Gamow’s boat and whiskey bottle puzzle intended to demonstrate the importance of relative frames of reference, although Kordemsky doesn’t tie his version to special relativity.

When I was looking over the customer reviews for The Moscow Puzzles on the amazon.com page, I was struck by just how shallow the negative comments were. Most of the readers gave the book 5 stars, but there was a small number of people that gave it one star, either because they were math teachers unable to figure out how to steal the problems for use in their own classes, or because some of the puzzles used props (like coins, dominoes, or cardboard cutout figures) that they were too lazy to gather together, or make themselves. And, yes, many of the puzzles use old-fashioned or cultural-specific references (e.g. – steam locomotives, or rebuilding the country after WW II), which would need to be updated or altered to be relatable for modern American children. But still, those complaints are awfully petty. Anyway, if you’re an adult and you like learning, then these puzzles are still fun pastimes. Additionally, yeah, not all puzzles are strictly paper and pencil things. You can treat them as games, with boards and playing pieces, and even play against friends or family members. Or, cut out some of the figures and give them to children to play with. Mathematics does have that branch called “game theory,” after all.

(Back cover.)

Towards the end of the book, I did start skipping over puzzles that I had less interest in, and my accuracy rate plummeted below 1 out of 6 puzzles. Especially for the dominoes, and memorization tricks puzzles. But, things picked back up when I got to the second to the last chapter, with instructions on how to create any sized magic squares. And with the last chapter, which was a hodge-podge of prime numbers, the Fibonacci series, and figurate numbers. With the figurate numbers, which relate to the regular polygons, Kordemsky hints at the canon ball stacking problem that appears in Gardner’s Colossal Mathematics book, without actually giving the problem itself. Really, though, the best part of reading these kinds of books is that they sometimes refer to other, older subjects that require that I look them up on wikipedia, often leading to stuff that I like that I didn’t know existed before. A case in point is a sums-of-the-squares problem Kordemsky mentions that appears on the blackboard of a 1895 painting by Russian Bogdanov-Belsky, titled “A Difficult Problem” (according to Dr. Albert Parry’s translation). The problem, and the painting itself are shown below. I think it’s interesting.

The next chapter on Symmetry focuses on Maurits Cornelis Escher (1898-1972). Gardner starts out by saying that there’s very little (as of that time) in the way of artwork based on mathematical concepts, ignoring the simple stuff using geometric patterns like cubes or triangles. The only real exception is with the works of M.C. Escher, who had said that he more closely associated with mathematicians than with his fellow artists. The rest of the chapter gives examples of the tie-ins to various branches of math with specific Escher prints (crystallography and the flying birds, symmetry and Day and Night, impossible objects and Belvedere). And, I mentioned Penrose’s “infinite staircase,” which was the inspiration for Ascending and Descending.

Gardner comments that Escher’s fascination with topology can be seen in Knots. The top two knots are mirror-image trefoils, with the top left being “made up of two long flat strips that intersect at right angles.” It was given a twist so it can either be a one-sided band that runs twice around the knot, or two distinct, intersecting Mobius bands. The large knot at the bottom is a quarter-turn Mobius band.

Comments:
I’ve long been fascinated with Escher’s works, and I once tried learning to draw by copying a few of them. More recently, I did kiri-e (the Japanese art of using cut paper) versions of two of his woodcuts.

When I first started going through the Gardner Colossal book, I made notes of things that I wanted to follow up on, or include in the blog entries. One of which was David Cope’sEMI (Experiments in Musical Intelligence). I’m not sure I’ll be able to do much with this program in the short-term, so I wanted to put the link in the blog now, in case anyone wants to check it out.

The Amazing Creations of Scott Kim is dedicated to Scott Kim, a puzzle designer and computer game designer. He studied under computer guru Donald Knuth at Stanford University. I mentioned before that there was a shopping complex in a revamped factory building in St. Paul, called Saint Anthony Main, that had a very eclectic bookstore that I’d visit while waiting for a table to open up at the Guadalaharry’s Tex-Mex restaurant. One of the books that I got there one day was Scott’s “Inversions,” (1981) which was my first introduction to his work on making text fonts (referred to as “ambigrams”) that could be read in more than one direction.

(Examples of Scott Kim’s ambigrams. All rights belong to their owners. Images used here for review purposes only.)

The chapters in Gardner’s book generally concentrate on one specific math concept, such as spherical ball packing, or infinity, but there are a few articles that focus on people, including Kim, Escher, Penrose and Conway. In this chapter, Gardner starts out talking about left-right and up-down mirroring of various text strings, and how a so-called “symmetrical calligraphy” can be achieved through a 180 degree rotation by stretching the letters as needed. The person he claims has taken the art to new heights is Scott Kim. Additionally, a few of Kim’s logic puzzles had been printed in Gardner’s SA column before, including the problem of placing chess knights on the corners of a hypercube.

(Scott’s invertible alphabet.)

Several pages discuss Kim’s ambigrams, and the fact that Omni devoted a page of their Sept. 1979 issue on them. Readers submitted their own ambigrams and Kim acted as the judge for the contest, with the prizewinners appearing in the April 1980 issue.

(More of Scott’s artwork.)

Eventually, Gardner switches to two different math problems Kim worked on. The first is the question of how many chess queen pieces you can put on a chessboard so that no queen attacks another, known as the Queens Problem (it’s a specific case of the more general mathematical chess problems). Kim switched this around to “what’s the maximum number of queens you can have on a board so that each one attacks exactly “n” other queens. If n=0, you have the original problem. For n=1, the max is 10.

(The Queen’s problem, solutions for n=1, n=2, and n=3.)

The second problem is one of how to fill space with snakes. A “snake” is a single connected chain of identical unit cubes that are joined at their faces such that each cube is attached face-to-face with exactly 2 other cubes, except at the cube at the end of the chain. A snake can twist in any direction possible as long as no two cubes abut the face of any other cube in the chain except with its immediate neighbors. A polycube snake can be finite, with two ends; finite and closed with no ends; infinite with one end; or, infinite and endless in both directions. What’s the smallest number of snakes to fill all space? For 2D space, the answer is “2 snakes.” As of the publication of this book, no one had yet found an answer for 3D space.

(Filling a space with 2 snakes.)

Puzzles:
With the above queen problem, the maximum number of queens to attack each other for n=4 is 20. Can you place the 20 queens on a regular chessboard for n=4?

(All rights belong to their owners. Images used here for review purposes only.)

The Godelian Puzzle Book: Puzzles, Paradoxes and Proofs, by Raymond Smullyan.
I first encountered the works of Raymond Smullyan in the book What is the Name of this Book? He started as a magician, then turned to mathematics with a degree from the University of Chicago in 1955, and a Ph.D. from Princeton in 1959. Several of his books are designed to teach the math principles of logic, and as such are great primers for this subject. The problem, as I see it, is that many of the readers that did get his puzzle books never actually learned anything from them, and were turned off by the apparent repetition of the puzzle set-ups. I can see that, having read two of his books back to back in the mid-80’s, and then not continuing on with his other titles. On the other hand, I liked his sense of humor, and I think he was a funny writer. Unfortunately, he passed away in February at age 97, and now I’ll never have a chance to meet him in person.

The current title was released in 2013, and was written as an introduction to Kurt Godel’s incompleteness theorems. The first few chapters set up the concepts of logic, infinity and paradoxes, many of which mirror the related chapters in the Martin Gardner Colossal Math Book. In fact, Gardner mentions Raymond in relation to Newcomb’s Paradox, and Raymond specifically talks about Martin’s interest in Newcomb’s paradox in this book. Again, everything is connected, and I love it

There’s one particular joke here that I want to repeat. “A certain man was in quest of immortality. He read many occult books on the subject, but none of them gave him any practical advice on how to become immortal. Then he heard of a certain great sage of the East who knew the true secret of immortality. It took him twelve years to find the sage, and when he did, he asked, “Is it really possible to become immortal?” The sage replied, “It is really quite easy, if you just do two things.” “And what are they?” the man asked quite eagerly. “First of all,” replied the sage, “from now on, you must always tell the truth. You must never make a false statement. That’s a small price to pay for immortality, isn’t it?” “Of course!” was the reply. “Secondly,” continued the sage, “just say ‘I will repeat this sentence tomorrow.’ If you do these two things, I guarantee you will live forever.” After thinking for a few minutes, the man said, “Of course if today I truthfully say that I will repeat this sentence tomorrow, then I will indeed say it again tomorrow, hence again the next day, and the next and the next and so on, but your solution is not very practical. How can I be sure of truthfully saying that I will repeat this sentence tomorrow if I don’t know for sure that I will be alive tomorrow? Your advice, though interesting, is simply not practical.”
“Oh,” said the sage, “You wanted a practical solution! No, I deal only in theory.”

The second half of the book is more textbook oriented, discussing the actual principles of mathematical logic theory and introducing symbolic logic. And, this is where my eyes started glazing over. If you actually want to sincerely learn the proofs, and you sit down with a pencil and paper and write out the symbols consistently to learn what they represent, coupled with the nomenclature of the proofs, then yes, this is a good, useful book. But there isn’t much in the way of real world examples to link the theory to practical application, so I got lost. I mean, I understood the basic concepts, and I could read some of the expressions in the proofs, but I’d never be able to explain the proofs to someone else at this point.

(Case in point.)

But, the main point is that in mathematics, mathematicians want to create specific types of math with clear rules that either hold, or don’t hold, and can be proven one way or another. Examples would be, in the world of natural numbers (positive integers starting with 1): that even numbers are divisible by 2; odd numbers leave a remainder of 1 when divided by 2; when adding two numbers the order doesn’t matter (2 + 3 = 3 + 2 = 5); if you add two numbers to get a third value, you can use either set when adding a third number (5 + 7 = 2 + 3 + 7 = 2 + 3 + 2 + 5 = 2 + 3 + 2 + 2 + 3); etc. So, we have these math guys trying to prove that the true statements are true, and that the false statements are false, and along comes Kurt Godel with a paper in 1931 proving that in any math system, there will be statements that you can not prove are either true OR false using the rules of that system. You CAN prove the statements one way or another if you analyze the one system using the next system up, but that becomes a meta-analysis, and that’s where logic theory steps in, with a vocabulary that groups statements, formulas, variables and sets together to generalize the system you want to look at. There is something of an example showing how this analysis works involving the Peano axioms as applied to the natural numbers, but that’s at the end of the book, is kind of short, and isn’t actually explained in simple words.

If you look at Smullyan’s puzzles, they often take the form of Knights and Knaves. The knights always tell the truth, the knaves always lie. In some cases, he introduces “certified,” where the status of the knight or knave has been established (or “uncertified” if it’s not established). And, we also get “magicians,” who can be either knights or knaves, but they don’t actually do anything different. Examples of these puzzles are:

1) You’re walking along a road and you get to a fork. One tine of the fork goes to the town of Knights, and the other goes to the town of Knaves. At the fork are two people, A and B, one is a Knight, the other is a Knave. What question can you ask one of them to determine which road goes where?

2) You meet three people (A, B, C), one of whom is a magician. They tell you:
A) B is not both a knave and a magician.
B) Either A is a knave or I am not a magician.
C) The magician is a knave.
Which types are all three people, and which one is the magician?

3) You find an island of certified and uncertified knights and knaves. You meet someone who says “I am not a certified knight.” Is he/she a knight or a knave, and certified or uncertified?

What I found ultimately useful in this book is that Raymond finally explains his puzzles’ tie-ins to logic theory.
1) Knights represent statements that are true.
2) Knaves represent statements that are false.
3) Certified means that the statement is provably true or false.
4) Uncertified means that the statement is undecidable. You can’t prove it is either true OR false.
5) “Magician” represents some property of the statement, e.g. – that natural whole numbers can be even, odd or prime.

So, you could have a statement regarding the natural numbers that is true about a specific property and still not be able to prove it. The rest of the book then goes on to develop the vocabulary and theorems for how to show Godel incompleteness for any system, including logic theory itself.

Overall, I like this book, and I now have a slightly better idea of what Smullyan was trying to demonstrate in his earlier puzzle books. I would have preferred more examples of unprovably true or false statements in number theory, because I do want a practical answer. Also, there are quite a few obvious typos in the book (misspellings, repeated or missing words, and the use of “proportional logic” instead of “propositional logic” on page 256) that were distracting but otherwise non-malicious. This is not a simple child-friendly introductory primer to Godelian theory. But, it could be good for math undergrads. Recommended if you like Smullyan’s other books.

—

Answer 1) If you ask A, say something like “If I asked B ‘does the left fork goes to the town of Knights?’, what would he say?”
Answer 2) A and B are both knights, C is a knave, and A is the magician.
Answer 3) Let’s call the person “A”. If A is a knave, they are lying and then HAVE to be an uncertified knight. But, knights don’t lie, so A can’t be a knave that is lying. However, if A IS a knave, certified or uncertified, then they are telling the truth in not being an uncertified knight, and knaves don’t tell the true. Alternatively, if A is a knight, who always tells the truth, then there is no option but for A to be an uncertified knight. (That is, A is a true statement that can’t be proved.)

Answers for rotation and reflection:
1) Oliver Lee, age 44, lives at 312 Main Street. He asked the city to give him the license plate 337-31770 for his car. Why?

Rotated 180 degrees, the plate reads “OLLIE LEE”.

2) A basket contains more than 6 eggs, which are either white or brown. If x is the number of white eggs and y is the number of brown eggs, then what values are needed such that the sum of x and y turned upside-down is the product of x and y? That is, how many eggs are there?

We’re now entering the section on Symmetry, starting with the chapter on Rotations and Reflections. First, from a geometrical viewpoint, something is considered to be symmetrical if it looks the same (its physical characteristics don’t changed) if you perform a “symmetry operation” on it. That is, if you hold a mirror up to the capital letter “A” at the halfway vertical point, it’s unchanged. “A” is said to have vertical symmetry. If you hold the mirror horizontally halfway up the letter “B”, you’ll see it has horizontal symmetry. The letter “S” is the same if you rotate it 180 degrees (twofold symmetry). “H,” “I,” “X” and “O” have all three symmetries. If the letter “X” is written as a cross, it’s unchanged if you rotate it 90 degrees (fourfold symmetry). The letter “O” is considered to be the “richest” in symmetry because it’s unchanged whatever you do to it.

The entire chapter is then dedicated to giving other examples of rotations and reflections. Gardner states that because the Earth is a sphere, with a center of gravity that pulls everything on it downward, living things have developed strong vertical symmetry, but very little horizontal or rotational symmetry. Things humans have created reflect this bias towards vertical symmetry – just look at how chairs, tables, dishes, cars, planes and office buildings are designed. Most normal representational art is similar in this respect. The exception is completely non-representational art, and in some cases museum curators have hung abstract paintings upside-down and no one noticed for months. Can’t do that with the Mona Lisa without realizing the mistake pretty quickly.

Pictures that look one way when right-side up, and something else when rotated have been popular tricks for political cartoonists since the 1800’s. A couple more recent examples were used by Life magazine. In the Sept. 18, 1950 issue, a reproduced Italian poster had the face of Garibaldi, and when turned over showed the face of Stalin. The back cover of the Nov. 23, 1953 issue showed an Indian brave inspecting a stalk of corn, and upside-down was a man looking hungrily at an open can of corn. Gardner mentions Peter Newell (1862-1924), who published 2 children’s books of color plates of scenes that transform when rotated: Topsys and Turvys and Topsys and Turvys Number 2. And there’s !OHO!, by Rex Whistler.

(Image from Amazon UK, used for review purposes only.)

Probably one of the more impressive displays of this kind of art is Gustav Verbeek’sThe Upside-Downs of Little Lady Lovekins and Old Man Muffaloo, a weekly strip that ran from 1903 to 1905, for a total of 64 strips. You read the first 6 panels normally, then turn the page over and read the same panels again to finish the story. GoComics carries the reprints now.

Salvador Dali illustrated Maurice Sandoz’s The Maze, which supposedly has several plates where turning the image causes it to look like something else, but I can’t find any example artwork on the net showing this.

Finally, a couple puzzles.
Oliver Lee, age 44, lives at 312 Main Street. He asked the city to give him the license plate 337-31770 for his car. Why?

A basket contains more than 6 eggs, which are either white or brown. If x is the number of white eggs and y is the number of brown eggs, then what values are needed such that the sum of x and y turned upside-down is the product of x and y? That is, how many eggs are there?

(All rights belong to their owners. Images used here for review purposes only.)

Penn and Teller’s Magic and Mystery Tour (2003)
I’ve seen bits and pieces of this DVD on youtube, and it really is better to watch the DVD itself. This was a 3-part TV mini-series directed and produced by the Canadian Broadcasting Corporation, along with Channel Four Film. The producers had wanted to do a piece on street magic around the world, and they asked Penn and Teller to act as hosts, as well as to actually visit other countries and interview the magicians they met. The countries selected were: China, Egypt and India. Each episode runs about 50 minutes, and consists of local color footage, footage of various street magicians performing, a couple of interviews, and Penn and Tell performing a couple tricks for the street magicians. The DVD extras section has video that didn’t make it into the shows. For China, that was Penn leading the village kids in a cheer (“Penn and Teller Rule”); an Egyptian that eats razor blades and glass; and an Indian that does a bullet catching routine.

China was kind of a special case, because the People’s government outlawed street performances, so most of the performers were working either in tea houses, failing government-run amusement parks, or small theaters. I’ve seen the one guy that does the mask switching dance here in Kagoshima, and I loved watching him again on the video. But, the rest of the Chinese performers were great, too. The Egyptian acts alternated between the cups and balls, and snake swallowing. I liked the primary cups and balls guy, but Penn’s commentary for everything happening in Egypt struck me as unnecessarily shallow. The Indian magicians, on the other hand, have absolutely mastered the geek act, cutting off bits of their children, killing them and then bringing them back to life. Disturbing, in a way. But, still a great DVD to get if you like Penn and Teller.

On a side note, I got this DVD as part of a birthday gift, shipped to me in Japan. When I received it, the package was neatly wrapped in gift paper, but it rattled when I picked it up. Removing the wrapping showed that it was indeed a DVD, but generally when you shake DVDs they don’t rattle like they’re in multiple pieces. At least, not if the idea is to put the pieces in a DVD player and actually play them. So, I was a bit hesitant to open the clamshell, but it was ok. The clamshell itself was so old that it had gotten brittle and was self-destructing. I took this to be a form of commentary on the DVD itself.